Determining mode I cohesive law of Pinus pinaster by coupling double cantilever beam test with digital image correlation

J. Xavier et alii, Frattura ed Integrità Strutturale, 31 (2015) 13-22; DOI: 10.3221/IGF-ESIS.31.02 Determining mode I cohesive law of Pinus pinaster by coupling double cantilever beam test with digital image correlation J. Xavier Centre for the Research and Technology of Agro-Environmental and Biological Sciences, CITAB, University of Trás-os-Montes and Alto Douro, UTAD, Quinta de Prados, 5000-801 Vila Real, Portugal. INEGI, FEUP, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal. M. Oliveira Centre for the Research and Technology of Agro-Environmental and Biological Sciences, CITAB, University of Trás-os-Montes and Alto Douro, UTAD, Quinta de Prados, 5000-801 Vila Real, Portugal. Instituto Superior Politécnico de Viseu, Departamento de Engenharia de Madeiras, Viseu, Portugal J.J.L. Morais Centre for the Research and Technology of Agro-Environmental and Biological Sciences, CITAB, University of Trás-os-Montes and Alto Douro, UTAD, Quinta de Prados, 5000-801 Vila Real, Portugal. M.F.S.F. de Moura INEGI, FEUP, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal. ABSTRACT. The direct identification of the cohesive law in pure mode I of Pinus pinaster is addressed. The approach couples the double cantilever beam (DCB) test with digital image correlation (DIC). Wooden beam specimens loaded in the radial-longitudinal (RL) fracture propagation system are used. The strain energy release rate in mode I ( GI ) is uniquely determined from the load-displacement ( P   ) curve by means of the compliance-based beam method (CBBM). This method relies on the concept of equivalent elastic crack length ( a eq ) and therefore does not require the monitoring of crack propagation during test. The crack tip opening displacement in mode I  w I  is determined from the displacement field at the initial crack tip. The cohesive law in mode I ( I  w I ) is then identified by numerical differentiation of the GI  w I relationship. Moreover, the proposed procedure is validated by finite element analyses including cohesive zone modelling. It is concluded that the proposed data reduction scheme is adequate for assessing the cohesive law in pure mode I of P. pinaster. KEYWORDS. Wood; Mode I fracture mechanics; Double cantilever beam test; Cohesive law; Digital image correlation. 13 J. Xavier et alii, Frattura ed Integrità Strutturale, 31 (2015) 13-22; DOI: 10.3221/IGF-ESIS.31.02 INTRODUCTION W ood is a hierarchical, anisotropic and heterogeneous composite material formed by trees. Recently, green composites based on lignocellulosic fibres and forest-based resources have attracted increasing interest in both research and market [1, 2]. Moreover, in a policy of sustainability, wood and wood products are increasingly used nowadays, for instance, in structural and semi-structural applications [3]. However, for a better and efficient utilisation of wood material, several issues must be further investigated. One fundamental aspect concerns the fracture mechanical behaviour of wood. Relatively extensive fracture process zones (FPZ) are observed in wood due to fibre bridging and micro-cracking ahead of the crack tip [4]. However, the microstructural mechanisms in wood fracture are usually confined to a region of reduced thickness [5]. Therefore, at the macroscopic scale, the wood behaviour in the FPZ can be conveniently described through a phenomenological constitutive cohesive law [6, 7]. In order to obtain the cohesive law, one approach consists in minimising an objective function quantifying the difference between numerical and experimental load-displacement ( P   ) curves by inverse analysis, assuming a given shape of the softening law. This approach, however, is semi-empirical and does not guarantee the uniqueness of the solution. Notwithstanding, it has been shown that the inverse identification of cohesive laws provide good agreement between experimental and numerical finite element simulations [7, 8]. Instead, a direct method for evaluating the cohesive law can be proposed based on independent determination of strain energy release rate and crack tip opening displacement (CTOD) [9]. The advantages of this approach are: (i) the shape of the cohesive law does not need to be assumed a priori; (ii) the cohesive law is determined based on local measurements. In this work, a direct identification of the cohesive law in mode I of P. pinaster was investigated by coupling the double cantilever beam (DCB) test with digital image correlation (DIC). Specimens oriented in the radial-longitudinal (RL) propagation system were used. The strain energy release rate in mode I ( GI ) was explicitly determined from the P   curve by means of the compliance-based beam method (CBBM). This data reduction scheme is based on the concept of equivalent elastic crack length ( a eq ) and, therefore, does not require the measurement of the crack length during test. An independent evaluation of CTOD in mode I ( w I ) was determined from displacement fields at the initial crack tip. The direct differentiation of the GI  w I curve and the reconstruction of the  I  w I cohesive law, by means of least-squares regression using a continuous approximation function, were addressed. The proposed procedure was also validated by finite element simulations including cohesive zone modelling. Figure 1: Schematic representation of the DCB test ( 2h = 20 mm, L1 = 300 mm, L = 280 mm, B = 20 mm and a 0 = 100 mm). DATA REDUCTION T 14 he DCB test is schematically shown in Fig. 1. The specimen is a L1  2h  B mm3 rectangular beam. The resistance curve (R–curve) can then be determined from the Irwin-Kies equation GI  P 2 dC 2 B da (1) J. Xavier et alii, Frattura ed Integrità Strutturale, 31 (2015) 13-22; DOI: 10.3221/IGF-ESIS.31.02 in which C   / P is the compliance. From the Timoshenko beam theory and Castigliano theorem an expression for the compliance of the DCB specimen can be obtained. This equation can be solved for the flexural modulus ( E f ) using an initial compliance (C 0 ) and the corrected initial crack length ( a 0   ) as [4] 1 12  a 0     8  a 0    3  E f  C0   5BhGLR  Bh 3  (2) where  accounts for root rotation effects and can be determined from finite element analysis [4], and GLR is the shear moduli of the material. The CBBM is based on a eq , which is considered to account for the FPZ effect at the crack tip as given by: a eq  a    a FPZ [7, 10]. Finally, the application of CBBM to the DCB test yield the following expression for the strain energy release rate in mode I (resistance or R–curve) [4] GI  2 6P 2  2a eq 1    2  2 B h  E f h 5GLR  (3) It is worth noting that this procedure is less sensitive to experimental errors. This is supported by the fact that the measurement of the crack length during the fracture test is not required. Besides, the inherent elastic properties variability among specimens is taken into accoun (...truncated)